A few thoughts here...
1- When the provider declined the requested time, saying "sorry I'm not available then" or "I have another appointment" should have been sufficient. Some folks are prone to oversharing. We don't need to know whether he's at the gym, doing laundry, seeing another client, visiting his mom, etc.
2- The provider should have proposed an alternate time. "I can't meet you at 2pm, but how about after 4pm?" If he didn't then take that as a no.
3- We as clients shouldn't get upset when someone isn't available the exact time we want them. I would try to have a range of time in my request ("Are you available between 1pm-6pm?")